February 6: Jesse Han (McMaster University), "Strong conceptual completeness for \omega-categorical theories"
Abstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is really an imaginary sort of our theory?

In the '80s, Michael Makkai proved that the answer to our question is yes if and only if our given process is compatible with all ultraproducts and all "formal comparison maps" between them (generalizing e.g. the diagonal embedding into an ultrapower). This is known as /strong conceptual completeness/; formally, the statement is that the category Def(T) of definable sets can be reconstructed up to bi-interpretability as the category of "ultrafunctors" Mod(T) \to Set.

\omega-categorical structures, having few definable sets, are exceptionally simple to understand, and in fact are determined up to bi-interpretability by the action of their automorphism groups. Any general framework which reconstructs theories from their categories of models should therefore be considerably simplified for \omega-categorical theories.

Indeed, we show:

1. If T is ω-categorical, then X : Mod(T) → Set is definable, i.e. isomorphic to (M \mapsto ψ(M)) for some formula ψ ∈ T, if and only if X preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when T is ω-categorical.

2. This definability criterion fails if we remove the ω-categoricity assumption. We construct examples of theories and non-definable functors Mod(T) \to Set exhibiting this.

Past talks:
November 28: Gabe Conant (University of Notre Dame), "A group version of stable regularity"
Abstract: Given an abelian group G and a subset A of G, one can construct a graph on G in which distinct elements x,y in G are connected if x+y is in A. If this graph is stable, then work of Malliaris and Shelah implies that it satisfies a strong form of Szemeredi's regularity lemma (and this has nothing to do with groups). A corollary of recent work of Terry and Wolf is that if G is a finite dimensional vector space over a prime field, then the regular partition of such a stable graph can be obtained using cosets of a subgroup. This motivates a statement of ``coset regularity" for subsets A of arbitrary finite groups G, such that "xy in A" is a stable binary relation. We prove this statement using local stable group theory and an ultraproduct construction. Joint with A. Pillay and C. Terry.